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On the Total Curvature and Betti Numbers of Complex Projective Manifolds University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2018 On The Total Curvature And Betti Numbers Of Complex Projective Manifolds Joseph Ansel Hoisington University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons Recommended Citation Hoisington, Joseph Ansel, "On The Total Curvature And Betti Numbers Of Complex Projective Manifolds" (2018). Publicly Accessible Penn Dissertations. 2791. https://repository.upenn.edu/edissertations/2791 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/2791 For more information, please contact [email protected]. On The Total Curvature And Betti Numbers Of Complex Projective Manifolds Abstract We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Christopher B. Croke Keywords Betti Number Estimates, Chern-Lashof Theorems, Complex Projective Manifolds, Total Curvature Subject Categories Mathematics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/2791 ON THE TOTAL CURVATURE AND BETTI NUMBERS OF COMPLEX PROJECTIVE MANIFOLDS Joseph Ansel Hoisington A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2018 Supervisor of Dissertation Christopher Croke, Professor of Mathematics Graduate Group Chairperson Wolfgang Ziller, Professor of Mathematics Dissertation Committee: Herman Gluck, Professor of Mathematics Wolfgang Ziller, Professor of Mathematics Acknowledgments I would like to express my sincere gratitude to Christopher Croke, my advisor, for his mentorship, encouragement and support, and for introducing me to the mathematics that led to these results. I would also like to thank Herman Gluck, Peter McGrath, Tony Pantev, Brian Weber and Wolfgang Ziller for many helpful and enjoyable conversations. ii ABSTRACT ON THE TOTAL CURVATURE AND BETTI NUMBERS OF COMPLEX PROJECTIVE MANIFOLDS Joseph Ansel Hoisington Christopher Croke We prove an inequality between the sum of the Betti numbers of a complex pro- jective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical the- orems of Chern and Lashof to complex projective space. iii Contents 1 Introduction 1 2 The Chern-Lashof Theorems and Their Spherical Formulations 9 3 Total Curvature and the Betti Numbers of Complex Projective Manifolds 30 4 Complex Projective Manifolds with Minimal Total Curvature 51 5 Total Curvature and the Geometry of Complex Projective Hyper- surfaces 58 iv Chapter 1 Introduction The central results of this paper are an inequality between the total curvature of a complex projective manifold and its Betti numbers and a characterization of the complex projective manifolds whose total curvature is minimal. We will prove: Theorem 1.1. Let M be a compact complex manifold, of complex dimension m, holomorphically immersed in complex projective space, and let T (M) be its total absolute curvature, as in Proposition 1.2 and Definition 1.4 below: 2m P m+1 A. Let βi be the Betti numbers of M with real coefficients. Then βi ≤ ( 2 )T (M). i=0 In particular, T (M) ≥ 2. B. If T (M) < 4, then in fact T (M) = 2. This occurs precisely if M is a linearly embedded complex projective subspace. The foundation for these results is a classical family of theorems that were 1 proved by Chern and Lashof. The definition of total absolute curvature for com- plex projective manifolds is based on an invariant which they defined, originally for submanifolds of Euclidean space, in [CL57] and [CL58]. We will define the total ab- solute curvature of a complex projective manifold in Definition 1.4 below. However, we will prove that the total absolute curvature of a complex projective manifold has the following geometric meaning: Proposition 1.2. Let M be a compact complex manifold holomorphically immersed π π < 2 in complex projective space. Let ν M be its normal disk bundle of radius 2 , and ? < π N let Exp denote the normal exponential map from ν 2 M to CP . Let T (M) be its total absolute curvature. Then: Z Z 2 ? 2 ? −1 < π N T (M) = jdet(dExp )jdV ol 2 = ](Exp ) (q)dV ol P : V ol(CP N ) ν M V ol(CP N ) C < π N ν 2 M CP π In the first expression, we integrate over the normal disk bundle of radius 2 π N ? −1 because 2 is the diameter of CP . In the latter, ](Exp ) (q) denotes the pre- image count via Exp? for a point q in CP N . Proposition 1.2 is equivalent to the meaning of Chern and Lashof's invariant, which they defined for a submanifold of Euclidean space as follows: Z 1 T (M) = det(A ) dV ol 1 : (1.1) V ol(SN−1) j ~u j ν M ν1M 2 1 In this definition, ν M is the unit normal bundle of the immersion and A~u is the second fundamental form of the normal vector ~u. Dividing by V ol(SN−1) ensures that T (M) is the same whether we regard M as a submanifold of RN , or of a 0 higher-dimensional space RN+N containing RN as a linear subspace. Chern and Lashof then proved: Theorem 1.3 (Chern-Lashof Theorems, [CL57, CL58]). Let M be a closed manifold of dimension n immersed in Euclidean space: A. (First Chern-Lashof Theorem) Let βi the Betti numbers of M, with coefficients n P in the integers or any field. Then βi ≤ T (M). In particular, T (M) ≥ 2. i=0 B. (Second Chern-Lashof Theorem) If T (M) < 3, then M is homeomorphic to the n-sphere. C. (Third Chern-Lashof Theorem) T (M) = 2 precisely if M is a convex hyper- surface in an (n + 1)-dimensional linear subspace of RN . In the introduction to their first paper on total curvature, [CL57], Chern and Lashof cite the theorems of Fenchel and F´ary-Milnor,in [Fe29] and [F´a49,Mi50], as motivation for their results. Fenchel's theorem states that a smooth closed curve γ in R3 has total curvature at least 2π, with equality precisely for plane convex curves. The Chern-Lashof theorems can be understood as a far-reaching general- ization of Fenchel's theorem, to compact Euclidean submanifolds of any dimension and codimension. The F´ary-Milnortheorem states that if γ has total curvature at 3 most 4π, twice the minimum in Fenchel's theorem, then it is unknotted. Part B of Theorem 1.1 gives a similar statement for complex projective manifolds. The definition of total absolute curvature for complex projective manifolds is an adaptation of Chern and Lashof's invariant: Definition 1.4 (Total Absolute Curvature of Complex Projective Manifolds). Let M be a complex manifold, of complex dimension m, holomorphically immersed in the complex projective space CP N . Its total absolute curvature T (M) is defined to be: Z (2N−2m−1) 2 sin r sin r π T (M) = jdet( cos(r)IdTpM − A~v)j cos(r) dV < ; V ol(CP N ) r r ν 2 M < π ν 2 M π π < 2 where ν M is the normal disk bundle of radius 2 , A~v is the second fundamental form of the normal vector ~v, r is its norm, and IdTpM : TpM ! TpM is the identity transformation of the tangent space to M at its base point p. Dividing by the volume of CP N in Definition 1.4 ensures that T (M) is the same whether we regard M as immersed in CP N , or in a higher-dimensional space 0 CP N+N containing CP N as a linear subvariety. The extra factor of 2 in Definition 1.4 will be explained in Proposition 3.1 and Remark 4.2. When we need to dis- tinguish the invariant defined for smooth submanifolds of Euclidean space in (1.1) from the invariant for complex projective manifolds in Definition 1.4, we will write the first as TRN (M) and the second as TCP N (M). In Definition 2.2, we will give a 4 formula for the total absolute curvature of smooth submanifolds of spheres, which we denote TSN (M). Chern and Lashof's invariant depends on the extrinsic geometry of a submani- fold of Euclidean space. However, Calabi proved in [Ca53] that if a K¨ahlermanifold (with a fixed metric) admits a holomorphic isometric immersion into complex pro- jective space, even locally, then any two holomorphic isometric immersions of this manifold into complex projective space are congruent by a holomorphic isometry of the ambient space. It follows that the total absolute curvature of a complex projec- tive manifold is actually part of its intrinsic geometry. It would thus be interesting to find a completely intrinsic representation of the total absolute curvature of a complex projective manifold. We will do so for curves in the complex projective plane: Theorem 1.5. Let Σ be a smooth curve in CP 2, with K the sectional curvature of its projectively induced metric. Then: Z 1 (K − 4)2 + 4 T (Σ) = dA : π 6 − K Σ Σ Note that if Σ is as above, then its sectional curvature is bounded above by 4, the holomorphic sectional curvature of the ambient space. This implies that the integrand in Theorem 1.5 is well-defined. Theorem 1.5 is a corollary of a more 5 general result about the total absolute curvature of complex projective hypersur- faces which we will state and prove in Theorem 5.4.
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